3.3.5 Radiative stars
• Repeat for mass continuity, energy generation and radiative energy transport to get 4 simultaneous eqs in the four indices α...
• Determinant of matrix in LHS is:
Denotes diffusive radiative transfer
in energy transport equation; Drad ≠ 0. AS 4013 Stellar Physics
3.3.6 Solutions
• Solutions of matrix equation are:
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Page ‹#› 3.3.7 Convective stars
• Energy transport by fully adiabatic convection gives:
• Matrix equation becomes:
• Slightly simpler determinant and solutions.
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3.3.8 Upper-main sequence stars • Dominant opacity source is electron scattering which is independent of ρ, T so n = s = 0. • Energy source is thermonuclear CNO cycle for which λ = 1 and ν ~ 15.
• Equation of state: ideal gas (P ∝ ρT) so χρ=χΤ=1. • Substitute these values in equations for radiative stars to get:
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Page ‹#› Upper MS stars
• Hence for M > 1 MSun:
• Not too bad: recall that L ∝ M3.6 for high-mass stars in binaries. • Average density falls, temperature rises with increasing mass. • But we’ve made some major approximations and haven’t solved any DEs!
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Interlude: The Virial Theorem
• Whole system is in equilibrium if hydrostatic equilibrium is satisfied at all r. • Hence get relation between average internal pressure and gravitational PE of whole system.
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Page ‹#› Average internal pressure
Volume-averaged Gravitational PE pressure of system Volume of system
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Pressure: nonrelativistic gas
Fundamentally important result! Means that tightly bound systems in hydrostatic equilibrium have high particle KE, i.e. they’re HOT.
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Page ‹#› Pressure: ultra-relativistic gas
• For a gas of ultra-relativistic particles:
• As UR limit is approached, binding energy decreases and system is easily disrupted.
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Common ultra-relativistic gases
• Radiation-pressure dominated: • Pressure from photons, UR by definition! – e.g. supermassive stars.
• Degenerate electron-pressure dominated: • If electron temperature becomes very high – e.g. massive white dwarfs.
• Both to be discussed in more detail later.
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